Abstract. This will be a partial introduction to the subject I first wrote about in 1995 in the monograph with the similar title (see my website). A shorter description is in the (published) 1998 paper no. 68 in my CV (on the website; also downloadable). There are more things on FOLDS on my website, and away from it as well. For instance, the MSC thesis of Christian Boudreault, which is relevant to the specific things I want to say, I still have to put on my website. Although my familiarity with the literature tends to be spotty, I have the long-standing and strong impression that FOLDS and FOLDS equivalences have generated little interest either in the logical or in the categorical community: let this be a warning for you. After generalities, I will concentrate on countable structures, the family of analytic equivalence relations given by FOLDS equivalences, and logical connections, e.g. a generalization of Scott's isomorphism theorem on countable structures.